Boundedness criterion for integral operators on the fractional Fock–Sobolev spaces

Abstract

We provide a boundedness criterion for the integral operator $S_{\varphi }$ on the fractional Fock–Sobolev space $F^{s,2}({{\mathbb {C}}}^n)$, $s\ge 0$, where $S_{\varphi }$ (introduced by Zhu [18]) is given by

$$\begin{aligned} S_{\varphi }F(z):= \int _{{\mathbb {C}}^n} F(w) e^{z \cdot \bar{w}} \varphi (z- \bar{w}) d\lambda (w) \end{aligned}$$

with $\varphi$ in the Fock space $F^2({{\mathbb {C}}^n})$ and $d\lambda (w): = \pi ^{-n} e^{-|w|^2} dw$ the Gaussian measure on the complex space ${\mathbb {C}}^{n}$. This extends the recent result in Cao et al. (Adv Math 363: 107001, 33 pp, 2020). The main approach is to develop multipliers on the fractional Hermite–Sobolev space $W_H^{s,2}({{\mathbb {R}}}^n)$.